When a nuclear reactor has been shut down, and nuclear fission is not occurring at a large scale, the major source of heat production will be due to the beta decay of these fission fragments. For this reason, at the moment of reactor shutdown, decay heat will be about 7% of the previous core power if the reactor has had a long and steady power history. About 1 hour after shutdown, the decay heat will be about 1.5% of the previous core power. After a day, the decay heat falls to 0.4%, and after a week it will be only 0.2%. The decay heat production rate will continue to slowly decrease over time; the decay curve depends upon the proportions of the various fission products in the core and upon their respective half-lives.

Sounds like they should cool down pretty quickly, I thought. I just hadn't realised how very powerful these beasts are. Fukushima reactor 1 (the one that went bang first) generates 439MW of electricity, and let's say it's 45% efficient (I gather that's on the high side, but it makes the mental arithmetic easier) - in other words only 45% of the heat produced gets turned into electricity, and our reactor is producing 1GW of heat.

"For this reason, at the moment of reactor shutdown, decay heat will be about 7% of the previous core power" - blimey, that's 70 megawatts. You'll burn your hand on that for sure !

"About 1 hour after shutdown, the decay heat will be about 1.5% of the previous core power." - it was I gather about 1 hour after shutdown that the tsunami struck and knocked out the cooling. Power would be 15 megawatts - or the same heat as 150,000 hundred-watt bulbs, or a 15,000-bar electric fire. Still pretty warm - you can see why the cooling would be missed.

"After a day, the decay heat falls to 0.4%" - 4 megawatts, not by any means cool.

"and after a week it will be only 0.2%" - "only" 2 million watts after a week !

Hmm. I don't know how much fuel is in the reactor, but at present, 3 days after shutdown, you've got say 3 megawatts of heat still being produced - and presumably, given the hefty shielding around the core, not being lost easily.

Uranium at 25C has a specific heat capacity of 27.665 J·mol−1·K−1 - in other words, it takes 27 joules of energy to raise the temperature of 1 mole of uranium atoms (say 238 grams) by one degree. Uranium melts at 1132 degrees C.

A watt = 1 joule per second.

OK, given no heat losses and a starting temperature of 132C, how much uranium could you melt in a day with 3 megawatts ?

3000000 x 24 x 60 x 60 = 2.592 x 1011 joules in a day.

27,665 joules approx to raise 238 grams of uranium from 132C to melting point 1132C. Note that this assumes the heat capacity stays the same over that 1,000 degree temperature range. It probably doesn't, but we're just doing some rough approximations.

Hmm. Given no heat losses and only uranium to melt, that looks like more than 2,000 tonnes a day ! I think a reactor has more like 50 tonnes in it. My maths may be awry, but if it ain't it certainly looks as if sans cooling a meltdown is not only possible, but likely.

UPDATE - apparently (thanks dearieme) the fuel is uranium dioxide, which has a higher MP (2865C) and a higher heat capacity of (approx - see above, the capacity varies with temperature) 85 joules per mole per degree. 270 grams per mole, 85*2600/270 to raise 1g from 265C to melting point. That's about 8.18 * x 102 joules to melt a gram, 8.18 * x 108 joules for a metric tonne. One day's heat output at 3 Mw is still enough to melt about 250 tonnes of fuel.